Newspace parameters
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2299.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.3576074247\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 5x^{5} + 13x^{4} + 10x^{3} - 10x^{2} - 8x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - 3x^{6} - 5x^{5} + 13x^{4} + 10x^{3} - 10x^{2} - 8x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + 4\nu^{5} + 2\nu^{4} - 17\nu^{3} + \nu^{2} + 15\nu + 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 14\nu^{3} - 8\nu^{2} + 13\nu + 5 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 13\nu^{3} - 10\nu^{2} + 11\nu + 6 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 7\nu^{5} - 6\nu^{4} + 28\nu^{3} + 3\nu^{2} - 18\nu - 3 \)
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| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 7\nu^{5} - 6\nu^{4} + 28\nu^{3} + 4\nu^{2} - 20\nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} + 6\beta _1 + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 10\beta_{6} - 9\beta_{5} + 3\beta_{4} - 2\beta_{3} + \beta_{2} + 16\beta _1 + 12 \)
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| \(\nu^{5}\) | \(=\) |
\( 27\beta_{6} - 24\beta_{5} + 12\beta_{4} - 10\beta_{3} + 4\beta_{2} + 46\beta _1 + 29 \)
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| \(\nu^{6}\) | \(=\) |
\( 95\beta_{6} - 81\beta_{5} + 37\beta_{4} - 27\beta_{3} + 17\beta_{2} + 131\beta _1 + 94 \)
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Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.70920 | 2.51386 | 5.33975 | −3.29335 | −6.81053 | −3.29667 | −9.04803 | 3.31947 | 8.92234 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.85364 | 2.10496 | 1.43599 | 2.31001 | −3.90185 | −0.796889 | 1.04547 | 1.43087 | −4.28194 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.57445 | −2.87867 | 0.478885 | 2.13300 | 4.53232 | 2.65365 | 2.39492 | 5.28677 | −3.35829 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.16188 | 0.129605 | −0.650032 | −2.74837 | −0.150586 | 0.979019 | 3.07902 | −2.98320 | 3.19328 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.142265 | −2.70021 | −1.97976 | −0.670490 | −0.384147 | −2.08436 | −0.566183 | 4.29113 | −0.0953876 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.13565 | 1.06427 | −0.710302 | −2.32001 | 1.20864 | 3.27291 | −3.07795 | −1.86733 | −2.63472 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.02125 | −1.23381 | 2.08547 | 3.58921 | −2.49385 | −2.72766 | 0.172756 | −1.47771 | 7.25472 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2299.2.a.o | ✓ | 7 |
| 11.b | odd | 2 | 1 | 2299.2.a.s | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2299.2.a.o | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2299.2.a.s | yes | 7 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):
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\( T_{2}^{7} + 4T_{2}^{6} - 2T_{2}^{5} - 22T_{2}^{4} - 13T_{2}^{3} + 24T_{2}^{2} + 18T_{2} - 3 \)
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\( T_{3}^{7} + T_{3}^{6} - 14T_{3}^{5} - 8T_{3}^{4} + 58T_{3}^{3} + 8T_{3}^{2} - 56T_{3} + 7 \)
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\( T_{7}^{7} + 2T_{7}^{6} - 19T_{7}^{5} - 37T_{7}^{4} + 97T_{7}^{3} + 179T_{7}^{2} - 89T_{7} - 127 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 4 T^{6} + \cdots - 3 \)
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| $3$ |
\( T^{7} + T^{6} - 14 T^{5} + \cdots + 7 \)
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| $5$ |
\( T^{7} + T^{6} + \cdots - 249 \)
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| $7$ |
\( T^{7} + 2 T^{6} + \cdots - 127 \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} + 4 T^{6} + \cdots - 1759 \)
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| $17$ |
\( T^{7} + 4 T^{6} + \cdots + 1587 \)
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| $19$ |
\( (T - 1)^{7} \)
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| $23$ |
\( T^{7} + 5 T^{6} + \cdots - 7233 \)
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| $29$ |
\( T^{7} + 30 T^{6} + \cdots + 156873 \)
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| $31$ |
\( T^{7} - 3 T^{6} + \cdots + 10123 \)
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| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 4441 \)
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| $41$ |
\( T^{7} + 15 T^{6} + \cdots + 62787 \)
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| $43$ |
\( T^{7} - 14 T^{6} + \cdots + 96803 \)
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| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 92211 \)
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| $53$ |
\( T^{7} - 5 T^{6} + \cdots + 8193 \)
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| $59$ |
\( T^{7} + T^{6} + \cdots - 58737 \)
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| $61$ |
\( T^{7} + 25 T^{6} + \cdots + 2116799 \)
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| $67$ |
\( T^{7} - 5 T^{6} + \cdots - 79051 \)
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| $71$ |
\( T^{7} + 35 T^{6} + \cdots + 1078347 \)
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| $73$ |
\( T^{7} + 12 T^{6} + \cdots - 9277 \)
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| $79$ |
\( T^{7} + 14 T^{6} + \cdots - 27083 \)
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| $83$ |
\( T^{7} + 10 T^{6} + \cdots + 17541 \)
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| $89$ |
\( T^{7} - 24 T^{6} + \cdots - 128541 \)
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| $97$ |
\( T^{7} - 3 T^{6} + \cdots + 34757 \)
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